What do mathematicians mean by Navier Stokes existence and smoothness problem? I still don't know what mathematicians mean by Navier-Stokes existence and smoothness. Since there is a reward for proving it, it seems important to them. (in past several months I've read online articles on this topic). 
Physically, what do we get from such a proof? Is it possible that we don't have existence and uniqueness mathematically, but that our physics still "works" somehow? Also, is there a consensus on whether, based on physical intuition (we are modelling real things after all), there must be existence and uniqueness to the pure math problem?
 A: I'm far from expert and my advice is basically to read Terry Tao's piece linked in John Rennie's comment. (Read Terry Tao is always good advice.)  But I would like to make a couple of points.
Singularities are not impossible or unphysical
Mainly, I wanted to clear up the idea that, if we find smooth solutions that became singular in finite time that would somehow mean that "physics broke". And therefore such solutions must be physically unreasonable and silly toys for mathematicians. That was my first reaction when I heard of this question back in the day.
What this misses is that physically the Navier-Stokes equation is just an effective theory - at some finite scale the fluid is actually made up of molecules. Suppose we found a solution of the NS equations  where the turbulence got finer and finer until it became infinite in finite time. Physically that would be a situation where the turbulence got finer and finer until neighboring atoms stopped moving together at all. Then it would no longer make sense to talk about a fluid and we would have to evolve the system using Newtownian mechanics for the individual molecules. There is nothing unphysical about that - it's just interesting.
Effective theories have singularities all the time. It just tells you that your effective theory is actually sensitive to the fine details even though you thought they were irrelevant. A classic example is the Caustic in optics (the bright lines and sparkles you see in a reflection). Geometric optics tells you the intensity is infinite on a caustic - this just means that things geometric optics ignores such as interference and finite wavelength must be considered.
Proving the existence or non-existence of solutions is in and of itself a physically interesting fact.
The physical argument is the 2nd Law
That said there I believe there is a strong physical reason not to expect finite blowup and that is the 2nd Law of Thermodynamics (Tao obliquely refers to Maxwell's Demon) -perhaps this is what people are referring to. To have a blow-up (especially a stable blow up), energy has to be consistently focus in smaller and smaller regions of phase space - thus it seems like the fluid would be consistently de-randomizing itself. Which on the molecular level is hard to reconcile with the 2nd Law. Put another way, the reason we expect to have a hydrodynamic description in the first place is the 2nd Law. The system should locally relax to the highest entropy state given the conserved quantities of density, momentum and energy. Therefore a breakdown of NS theory is in some sense a breakdown of the 2nd Law, or at least or naive expectations thereof. (Having just typed all that I am pretty unsatisfied so I would be happy for a clarification or refutation.)
Mathematicians are people too
Finally, let us suppose we were physically certain that "generically" "non-crazy" data must have solutions for all time and the only real mathematical issue was finding the precise technical sense of "generically" and "non-crazy". Even then it is clear from TT's piece that one could not make progress without making great progress in understanding turbulence or non-linear PDE's. Both of which are issues of tremendous importance. Mathematicians' main task and physicists' main task is to understand the NS equations. It's just that once they do it that the mathematician's will go on to prove global regularity, and the physicists will go on to make better predictions.
A: The scientific cycle is something like experiment, model, predict, compare predictions to experiments, repeat. In the case of Navier Stokes, we do not know what all the predictions are; for example we don't know whether solutions are uniquely determined by initial conditions. Therefore we don't know how good the model is. Although you did not mention it, a third aspect of the problem is numerical methods. We are fairly confident in Navier Stokes, partly because numerical results agree reasonably with experiment. But that means there are at least 3 aspects to the problem: theory, numerics, and experiment. They are all related, all interesting, and I guess no two of them have been shown to be equivalent to each other.
A: Mathematician means:


*

*Is it possile to make a theoretical model to describe the statistics of a turbulent flow (in particular its internal structures) ?

*Also, under what conditions do smooth solutions to the NS-equations exist?
